Spaces of directed paths on pre-cubical sets II

Abstract

For a given pre-cubical set ( -set) K with two distinguished vertices 0, 1, we prove that the space P (K) 1 0 of d-paths on the geometric realization of K with source 0 and target 1 is homotopy equivalent to its subspace P t (K) 1 0 of tame d-paths. When K is the underlying -set of a Higher Dimensional Automaton A, tame d-paths on K represent step executions of A. Then, we define the cube chain category of K and prove that its nerve is weakly homotopy equivalent to P (K) 1 0 .

Strictifying and taming directed paths in Higher Dimensional Automata

“…Remark 38. By far more sophisticated homotopy theoretical methods, Ziemiański proved in Ziemiański (2020a) that T(X) x + x − is homotopy equivalent to the nerve of a Reedy category Ch(X) It seems to be necessary to replace the cube chains from this paper by the cube chains Ch(X) in Ziemiański's paper (Ziemiański, 2020a); those are generated by cubical maps from a wedge of cubes into X. Hyperplane sections in X should then be replaced by hyperplane sections in a wedge of cubes. In such a wedge of cubes (which is obviously both proper and non-self-linked), there is again a well-defined unit speed line segment between points on consecutive hyperplane sections.…”

“…It has been shown by Ziemiański (2017Ziemiański ( , 2020a) that the synchronisation request has no essential significance: The spaces of directed paths and of tame d-paths between two states are always homotopy equivalent. This has two consequences: On the one hand, one may, without global effects, relax the computational model and allow quite general parallel compositions.…”

“…In that line of argument, posets enter the scene in a different manner: We divide the space of all executions (d-paths) into easy-to-analyse subspaces; for tame d-paths, for example, we simply fix a sequence of faces that they are kept in. Refinement is a partial order relation on these face sequences, and we will exploit the combinatorial/topological properties of the poset category of face sequences (called cube chains Ziemiański, 2017Ziemiański, , 2020a.…”

Configuration spaces and directed paths on the final precubical set

“…To prove the main theorem, we need to show that there is a homotopy equivalence P ( Zn ) 1 0 UConf(n, R 2 ). A method of calculating the homotopy type of the execution space of K is described in [14]. Namely, P (K) 1 0 is weakly homotopy equivalent to the nerve of the cube chain category Ch(K) of K. If K is non-self-linked, then P (K) 1 0 has the homotopy type of a CW-complex, so we obtain a genuine homotopy equivalence.…”

“…Namely, P (K) 1 0 is weakly homotopy equivalent to the nerve of the cube chain category Ch(K) of K. If K is non-self-linked, then P (K) 1 0 has the homotopy type of a CW-complex, so we obtain a genuine homotopy equivalence. Unfortunately, Zn is not non-self-linked for n > 1, so we cannot apply the methods of [14] directly. Instead, for a fixed set A having n elements, we construct a non-self-linked -set Y A with an action of the group Σ A of permutations of A, and a -map Y A → Zn .…”